Like English, German has three types of articles:
- in English: the (for all genders, both numbers and all cases)
- in German: der, die, das, des, dem, den (depending on the combination of grammatical gender, number and case)
The sky is blue.
Der Himmel ist blau.
- in English: a, an (depending on the pronunciation of the following word, but only for singular)
- in German: ein, eine, eines, einem, einen, einer (depending on the combination of gender and case, but also only for singular)
A car is approaching.
Ein Auto kommt näher.
the null article
- this is just another way to say "there is no article". When you explicitly want to mark a null article, you sometimes use the symbol "ø":
ø Walter drinks coffee.
ø Walter trinkt Kaffee.
The sentence from your example is:
Der1 Betrieb ohne ø2 Beachtung des3 Handbuchs kann ø4 Maschine, ø5 Umwelt und Sie selbst schädigen.
So here we have five articles in this sentence, two are definite articles (1 and 3) and three are null-articles (2, 4 and 5).
1 has to be a definite article, because we are talking about a certain usage/operation. This is exactly the same rule as in English. We are talking about "the operation", not about "an operation".
2 can be any of the three types. When selecting a non-null-article you have to select the right one according to gender, number and case. Here we have a female word in singular and accusative case. So it could be:
Der Betrieb ohne die Beachtung des Handbuchs.
Der Betrieb ohne eine Beachtung des Handbuchs.
Der Betrieb ohne ø Beachtung des Handbuchs.
Here in this case all three sentences mean the same and all three of them are correct. Just the second with the infinite article is bad style. All three mean the same because Beachtung (paying attention or following) is an abstract noun, which means, that there is only one Beachtung (you very rarely are talking about two or more Beachtungen), and so, when you talk about Beachtung, you are always talking about the same thing. So it makes no difference, if you say "die Beachtung" or just "ø Beachtung". "Eine Beachtung" would mean just any Beachtung, but since there is just one Beachtung, this is bad style.
In 3 you need an article to mark the genitive case. Without an article (i.e. when using the null article) or a pronoun you can't build the genitive case. But you still have the choice between definite and indefinite article:
... Beachtung des Handbuchs ... (definite article)
... Beachtung eines Handbuchs ... (indefinite article)
it is exactly:
... following the manual ... (definite article)
... following a manual ... (indefinite article)
Here we are talking about a certain manual, so we use the definite article.
4 and 5 are part of a list (together with "Sie selbst"), and as a rule of thumb you should use the same type of article for all members (but there might be reasons for not following this rule). "Sie selbst" (yourself) can't have an article, because it is not a noun, so this is a hint for not using any articles in the list. If you replace "Sie selbst" by a noun then I would prefer articles:
Der Betrieb ohne Beachtung des Handbuchs kann die Maschine, die Umwelt und das Personal schädigen.
The version with null-articles is also ok, but sounds a little bit incomplete to me. But as just said, it is ok too:
Der Betrieb ohne Beachtung des Handbuchs kann Maschine, Umwelt und Personal schädigen.
Using an indefinite article is not recommended here, because Umwelt (environment) is a singularetantum, i.e. a word that has no plural. There is only one Umwelt, and it is not abstract. If it was abstract, you could think of an indefinite article. This would be correct, but bad style. But Umwelt is not abstract. You can see it and you can touch it (at least parts of it). And for this reason you can not use an indefinite article. Only the definite article is possible, or the null article.
(The english word for Umwelt is environment, and there is a plural for environment in English which is environments like in: "The environments of those systems are different." If you use it this way, then "Umgebung" is the matching German word. This is the next environment. But "Umwelt" is everything that surrounds us, so there is only one Umwelt, but there can be many Umgebungen. But both is "environment" in English.)
It is thinkable, that we are not talking about a certain machine, but about just any machine. In this case you might use the indefinite article here. This is a case, where you can violate the rule of thumb I told you before:
Der Betrieb ohne Beachtung des Handbuchs kann eine Maschine, die Umwelt und Sie selbst schädigen.
But mixing the types of articles within a list is not very fine (although allowed), so you better find another solution.